In all of these cases, cocycles cc on XX with coefficients in AA may be modeled by spans of the form

X←∈FWX˜→A
\array{
X &\stackrel{\in FW}{\leftarrow}& \tilde X \to A
}

in the ordinary categoryCC, where the morphism on the left is taken from a special class of morphisms (for instance from the class of acyclic fibrations in the case that CC is a category of fibrant objects). In each case the relevant hom-set in the homotopy categoryH(X,A)=π0H(X,A)H(X,A) = \pi_0 \mathbf{H}(X,A) is given by the collection of cocycles module an equivalence relation given by coboundaries. In formulas

Terminology

In the existing literature on localizations, the spans X←∈FWX˜→X X \stackrel{\in FW}{\leftarrow} \tilde X \to X are often not called by a dedicated special term. On the other hand, in the existing literature that explicitly uses the term “cocycle”, often more pedestrian definitions are used and it is not made explicit that morphisms in a homotopy category are being represented.

Notice that this article chooses to work with the full structure of a model category but presents constructions for cocycles entirely analogous to and in fact inspired by those used in a category of fibrant objects or in one equipped with a calculus of fractions. The author emphasizes that he can give a definition where the left leg of the cocycle spans are not required to be acyclic fibrations, but can be any weak equivalences. But all this is just a technical question of how exactly to model a cocycle, not a question of principle of concept. For instance in this context every cocycle defined with respect to a weak equivalence over its domain is cohomologous to one defined with respect to an acyclic fibration over its domain.

One could take this as a suggestion to find a dedicated term for spans as above and call generally such a span an anamorphism. An anamorphism would be effectively the same as a cocycle, but the term morphism in it would amplify the nature of cocycles as morphisms.

In most cases the morphism ω:V•→Bnk\omega : V_\bullet \to \mathbf{B}^n k defined this way is already a morphism in the relevant (∞,1)-categoryHCh•\mathbf{H}_{Ch_\bullet} of chain complexes: this is modeled for instance by the projective model structure on chain complexes. In this every object is fibrant, and the cofibrant objects are those consisting of projective kk-modules. If we assume that all our modules are projective (for instance in the archetypical case that our modules are simply vector spaces), then ω:V•→Bnk\omega : V_\bullet \to \mathbf{B}^n k is a cocycle in HCh•\mathbf{H}_{Ch_\bullet} from the above abstract nonsense point of view. For its cohomology class we may write